Conjugate momentum in the hamiltonian

In summary, the conversation discusses the concept of conjugate momenta in the Hamiltonian, particularly in the presence of a magnetic field. The term H=\frac{1}{2m}(p-qA)^2 is mentioned and it is questioned whether p or p_c=p-qA represents the conjugate momentum. It is concluded that p is the conjugate/canonical momentum and p-eA is the mechanical momentum. The role of -i\hbar\nabla as the conjugate/canonical momentum operator is also clarified.
  • #1
moobox
2
0
Hey,

I just have a quick question that I haven't quite been able to find a definitive answer to, regarding conjugate momenta in the Hamiltonian.

Ok, so it regards the following term for the hamiltonian in a magnetic field:

[tex]H=\frac{1}{2m}(p-qA)^2 [/tex]

I'd like to ask whether [tex]p[/tex] is the conjugate momentum or if [tex]p_c=p-qA[/tex] is the conjugate momentum. As a guess, I would say that [tex]p_c=p-qA[/tex] is the conjugate momentum, as it seems to me that the hamiltonian should take into account the magnetic field. Would this then mean that the hamiltonian could be written as [tex]H=\frac{1}{2m}(p_c)^2 [/tex]

Also, very important, does [tex] -i\hbar\nabla[/tex] represent the canonical momentum operator or the classical/mechanical momentum operatpor?

Im sure the answers are around somewhere on the internet, but it strikes me that there are some conflicting statements and a tendency to just go "oh yeah, now we swap the canonical momentum, [tex]p[/tex] for mechanical momentum [tex]p[/tex]" and the like, so it would be nice to get a definitive answer.

Thanks for your help!
 
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  • #2
p is the conjugate/canonical momentum. p-eA is the mechanical momentum.
 
  • #3
Ah, brilliant! Thanks for your quick response!

So that would make [tex] i\hbar\nabla[/tex] the conjugate/canonical momentum operator?
 

1. What is conjugate momentum in the Hamiltonian?

Conjugate momentum in the Hamiltonian is a concept in classical mechanics that represents the momentum associated with a particular coordinate in a system. It is derived from the Hamiltonian function, which is a mathematical expression that describes the total energy of a system.

2. How is conjugate momentum related to the Hamiltonian?

Conjugate momentum and the Hamiltonian are closely related, as they both describe different aspects of a system's energy. The Hamiltonian function is a combination of the system's kinetic and potential energy, while the conjugate momentum represents the momentum associated with a specific coordinate in the system.

3. What is the formula for calculating conjugate momentum in the Hamiltonian?

The formula for calculating conjugate momentum in the Hamiltonian is p = ∂H/∂q, where p is the conjugate momentum, H is the Hamiltonian function, and q is the coordinate associated with the momentum. This formula is derived from Hamilton's equations of motion.

4. How does conjugate momentum affect the dynamics of a system?

Conjugate momentum plays a crucial role in determining the dynamics of a system. It is used to calculate the system's equations of motion, which describe how the system's position and momentum change over time. The value of the conjugate momentum at a particular point in time can also be used to predict the system's future behavior.

5. What are some applications of the concept of conjugate momentum in the Hamiltonian?

The concept of conjugate momentum in the Hamiltonian has various applications in physics and engineering. It is used in the study of classical mechanics, quantum mechanics, and statistical mechanics. It also has applications in fields such as celestial mechanics, electromagnetism, and fluid dynamics.

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